The membrane

Pin a few heights and the sheet relaxes to the Gaussian-process interpolant. A soap film, almost.

The smoother was a chain of beads in one dimension. Lift it into two. Stretch an elastic membrane over a square frame and push it up or down with a few posts. The sheet settles into the shape that stores the least energy. For gentle slopes that energy is the Dirichlet energy, a quadratic, so the resting sheet is again a Gaussian object.

$$ E[u] \;=\; \tfrac12 \int_\Omega |\nabla u|^2 \, dA, \qquad \nabla^2 u \;=\; 0 \ \text{ away from the posts.} $$

Minimising that energy makes the height harmonic between the posts. It obeys Laplace's equation, the two-dimensional reading of “a straight line between two pinned beads”. Fix the frame and a few interior points and the membrane relaxes to the one harmonic surface that meets them. That surface is the posterior mean of a Gaussian field conditioned on the pinned heights. Pinning a height here is conditioning, the move of the graph demo, now on a continuum rather than six beads.

Drag a post up and down and the whole sheet swells or dips around it, tallest at the post and decaying outward. Click the flat frame to add a post, and clear them to start over. The readout tracks how far the surface still is from exactly harmonic. The relaxation drives that to zero, so what you see is the exact interpolant, not a rough stand-in for it.

posts distance from harmonic drag a post to set its height

The membrane is the mechanical face of kriging and the thin-plate spline. Danie Krige introduced the statistical version in 1951 for valuing gold on the Witwatersrand, and Georges Matheron built the theory through the 1960s. Jean Duchon set out the thin-plate spline in 1976, a sheet that minimises bending energy rather than tension, which is the two-dimensional cousin of the smoother's second-difference penalty.

And a soap film, almost. A real film minimises its area, $\int\sqrt{1+|\nabla u|^2}\,dA$, not the quadratic $\tfrac12\int|\nabla u|^2$. The two agree only while the slopes stay gentle. Joseph Plateau catalogued the true minimal surfaces by dipping wire frames in the 1870s, and the existence of a film for any frame was settled by Jesse Douglas and Tibor Radó in 1930. Push the posts far enough and the sheet leaves the Gaussian world, the same way the Lasso leaves it at the kink.

The usual explanation from Dirichlet energy to Laplace

Fix the height $u$ on the frame and posts and minimise the Dirichlet energy over everything else. Perturb the surface by an admissible bump $\eta$ that vanishes wherever the height is held, and take the first variation:

$$ E[u] = \tfrac12\int_\Omega |\nabla u|^2\, dA, \qquad \frac{d}{d\varepsilon}\,E[u+\varepsilon\eta]\Big|_{\varepsilon=0} = \int_\Omega \nabla u\cdot\nabla\eta\, dA. $$

Green's identity moves the derivative off $\eta$. The boundary term drops because $\eta$ vanishes on the frame and posts:

$$ \int_\Omega \nabla u\cdot\nabla\eta\, dA = -\int_\Omega (\nabla^2 u)\,\eta\, dA. $$

At a minimum this vanishes for every admissible $\eta$. That forces the bracket to zero, which is Laplace's equation:

$$ \nabla^2 u = 0. $$

Discretise on the square grid. The five-point Laplacian sets each free node's second difference to zero, so the node equals the average of its four neighbours:

$$ u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1} - 4u_{i,j} = 0 \;\Longrightarrow\; u_{i,j} = \tfrac14\big(u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}\big). $$

That neighbour-average condition is the Gaussian-process conditional mean, and the relaxation loop drives the surface to satisfy it everywhere. Several lines of calculus, and you have the harmonic interpolant.

The physics proof the graph demo, now in 2D

This sheet is the graph demo laid out in two dimensions. Every free node is tied to its four neighbours by equal springs, so the centre-of-mass rule from the fusion page applies node by node:

$$ u_{i,j} = \tfrac14\big(u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}\big). $$

Each node wants to sit at the average of the four around it. Pin the frame and a few posts, release the sheet, and it relaxes until every node holds that balance. The surface that results is harmonic. It is the interpolant, found by letting go.